I did finally find the vacuum connection on the carb just shortly before Matt came over. It was plugged off with a very short piece of tube with a screw in the end. In dim light, it looked just like another adjustment screw...A weak battery will certainly not help with diagnosing the problem. I'm still thinking it's your timing/advance. Did you get the Vacuum advance connected?
I was looking at another 40 recently and it made me think of this thread. They had the vacuum advance connected to the valve on the intake manifold (in front of the carb riser) rather than connecting it to a ported vacuum connector on the carb. Not sure if this would be detrimental, but it seemed to be working in their application.
Do you still have a toyota intake manifold? and if so, can you connect your vacuum advance to that instead of nothing?
(hopefully the toyota fj40 carb gurus will chime in and school us on vacuum advance)
From the reading I've done, it sounded like I should have it disconnected and plugged when I adjust the timing, and reconnected when I actually run the car, so this is what I've been doing. Also, from what I've read, timing with the Weber should be around 12 degree advanced. The timing light I borrowed is just the basic variety, but it sounded like 12 degrees should be when the BB is just out of view out of the bottom of the window. Whenever I got it to come into view, the engine would start to die.

